1. Hanghang Tong, Brian Gallagher, Christos Faloutsos, Tina Eliassi-Rad
L L N L
Graph X-Ray: Fast Best-Effort Pattern Matching in Large Attributed Graphs
Hanghang Tong, Brian Gallagher,
Christos Faloutsos, Tina Eliassi-Rad
8/13/2007
KDD 2007, San Jose

2. Input
Output
Query Graph
Matching Subgraph
Attributed Data Graph

3. Terminology: ``Conform’’
First, We say the subgraph H_t conforms the query graph H_q, if we have all desired job titles and connection between them.
Matching Subgraph conforms
Query Graph

4. Terminology: ``Interception’’
Intermediate node
matching node
matching node
matching node
matching node
We allow the in-directed connection by introducing some extra nodes. For example, the connection between 12 and 4 is indirected. We refer this phenomena as interception, and the extra nodes, e.g. node 13 as intermediate node. And all remaining nodes as matching nodes, e.g. node 11 12,4 and 7.
Matching Subgraph
Query Graph
Path is an Interception

5. Terminology: ``Instantiate’’
Matching Subgraph Ht
Query Graph Hq
Whenever we have a matching subgraph H_t, we say H_t instantiates the query graph H_q. and the matching nodes in H_t instantiates the nodes in the query graph. for example, we say node 11 in H_t instantiates the SEC node in the query graph, and so on.
Node 11 instantiates SEC node
Ht instantiates Hq

6. Roadmap Introduction How to: Graph X-Ray Experimental Results
Problem Definition
Motivations
How to: Graph X-Ray
Experimental Results
Conclusion

7. Motivation: Why Not SQL?
Case 1: Exact match does not exist
Q: How to find approximate answer?
Case 2: Too many exact matches
Q: How to rank them?

8. Motivation: Why Not SQL? (Cont.)
Case 3: Exact match might be not the best answer
``Find CEO who has heavy contact with Accountant’’
Q: how to find right?
Exact match
1 direct connection
Inexact match
Many indirect connections

9. Motivation: Efficiency
Why Not Subgraph Isomorphism?
Polynomial for fixed # of pattern query
Q1: How to scale up linearly?
Q2: … and with a small slope?

10. Wish List G-Ray meets all! Effectiveness Efficiency
Both exact match & inexact Match
Ranking among multiple results
``Best’’ answer (proximity-based)
Efficiency
Scale linearly
Scale with small scope
G-Ray meets all!

11. Roadmap Introduction How to: Graph X-Ray Experimental Results
Problem Definition
Motivations
How to: Graph X-Ray
Experimental Results
Conclusion

12. Preliminary: Center-Piece Subgraph [Tong+]Q
Original Graph
Black: query nodes
CePS is meta opt. in G-Ray!

13. Preliminary: Augmented Graph
Data nodes
1,…13
Attribute nodes a
Footnote
Aug. Graph is crucial for computation!

14. G-Ray: quick overview (for loop )
Step 1: SF
Step 2: NE
Step 3: BR
Step 4: NE
Step 5: BR
Step 6: NE
Step 7: BR
Step 8: BR
SF: Seed-Finder
NE: Neighborhood
-Expander
BR: Bridge

15. Seed-Finder ( ) Q: How to instantiate SEC node? A: Footnote
`11’ is close to some un-known
data nodes for `CEO’ `Account.’
and `Manager’

16. Neighborhood-Expander ( )
Q: How to instantiate CEO node?
Step 1  Step 2? A:
Footnote:
Step 3  Step 4?
Step 5  Step 6?

17. Bridge ( ) ? Q: A: Prim-like Alg. Footnote To maximize
Step 6: NE
Step 7: BR ? Q:
A: Prim-like Alg.
To maximize
Should block node 11 and 7
Footnote
Connection subgraph, or one single path?

18. Roadmap Introduction How to: Graph X-Ray Experimental Results
Problem Definition
Motivation
How to: Graph X-Ray
Experimental Results
Conclusion
Now, let’s see some experimental results.

19. Experimental Results Datasets DBLP Node: author (315k)
Edge: co-authorship (1,800k)
Attribute: conference & year (13k)
KDD-2001, SIGMOD…
We use DBLP to construct an attributed graph, where the nodes are authors and attribute is conference and year. The edge is constructed from co-authorship relationship.

20. Effectiveness: star-query
Here is a star-query, we want to a star-shape group of co-authors, with one author coming from each of PODS, IAT and ISBMS. We see Dr. Phillips Yu is in the center and the rest matching authors being well known domain experts in each conf.
Query
Result

21. Effectiveness: line-query
And here is a line query, we want to find authors from 4 different conferences who cooperate in a line fashion.
Result

22. Effectiveness: loop-query
And this is a loop query.
Result

23. Efficiency Response Time # of Edges Scale linearly Small slope
3-5 Seconds
# of Edges
~2 M edges

24. Roadmap Introduction How to: Graph X-Ray Experimental Results
Problem Definition
Motivation
How to: Graph X-Ray
Experimental Results
Conclusion

25. Conclusion Graph X-Ray (G-Ray) More details in Poster Session
Best effort pattern match
in large attributed graphs
Scale linearly
with small slope
More details in Poster Session
Monday (tonight)
board number 8

26. G-Ray
X-Ray
Thank you!

27. Backup-slides

28. Proximity on Graph a.k.a relevance, closeness Multi-faceted1 4 3 2 5 6 7 9 10 8 11 12
a.k.a relevance, closeness
Multi-faceted
Punish long path
Edge weight
Now, I will introduce some key concepts behind G-Ray. Once we have these key concepts, the alg. itself is quite straight-forward.
So, the fist one is the proximity on the graph. How can we measure the proximity, or in other words, the relevance, the closeness, between two nodes on the graph.
Without going into the details, I want claim that random walk with restart is a good solution for this problem.
suppose
How to:
---- random walk with restart

29. Random walk with restart1 4 3 2 5 6 7 9 10 8 11 12
0.13
0.10
0.05
0.08
0.04
0.02
0.03
Node 4
Node 1
Node 2
Node 3
Node 5
Node 6
Node 7
Node 8
Node 9
Node 10
Node 11
Node 12
0.13
0.10
0.22
0.05
0.08
0.04
0.03
0.02
Nearby nodes, higher scores
Ranking vector
More red, more relevant

30. How to rank the results Our goodness function
Measure the proximity between any two matching nodes if they are required to be connected. (two-way)
Multiply them together
In G-Ray, we approximately optimize this goodness functions
If we have multiple matching subgraphs, we can rank them according to this goodness functions

31. How to rank the results Goodness = Prox (12, 4) x Prox (4, 12) x
matching node
matching node
matching node
matching node
Goodness = Prox (12, 4) x Prox (4, 12) x
Prox (7, 4) x Prox (4, 7) x
Prox (11, 7) x Prox (7, 11) x
Prox (12, 11) x Prox (11, 12)